Bond alternation, polarizability, and resonance detuning in methine dyes

We derive structure-property relationships for methine ("Brooker") dyes relating the color of the dye and its symmetric parents to its bond alternation in the ground state and also to the dipole properties associated with its low-lying charge-resonance (or charge-transfer) transition. We calibrate and test these relationships on an array of different protonation states of the green fluorescent protein chromophore motif (an asymmetric halochromic methine dye) and its symmetric parent dyes. The relationships rely on the assumption that the diabatic states that define the Platt model for methine dye color [J. R. Platt, J. Chem. Phys. 25, 80 (1956)] can also be distinguished by their single-double bond alternation and by their charge localization character. These assumptions are independent of the primary constraint that defines the diabatic states in the Platt model--specifically, the Brooker deviation rule for methine dyes [L. G. S. Brooker, Rev. Mod. Phys. 14, 275 (1942)]. Taking these assumptions, we show that the Platt model offers an alternate route to known structure-property relationships between the bond length alternation and the quadratic nonlinear polarizability β. We show also that the Platt model can be parameterized without the need for synthesis of the symmetric parents of a given dye, using the dipole data obtained through spectroscopic measurements. This suggests that the Platt model parameters may be used as independent variables in free-energy relationships for chromophores whose symmetric parents cannot be synthesized or chromophores strongly bound to biomolecular environments. The latter category includes several recently characterized biomolecular probe constructs. We illustrate these concepts by an analysis of previously reported electroabsorption and second-harmonic generation experiments on green fluorescent proteins.     

Finite equational bases for congruence modular varieties

In this paper it is proved that a variety generated by a finite algebraic system with finitely many operations is finitely axiomatizable, provided that the variety is congruence modular and residually small. This result is an extension to congruence modular varieties of a well known theorem for congruence distributive varieties, due to K. A. Baker. Also, under somewhat less restrictive hypotheses, (which are satisfied by finite groups and rings) it is proved that a finite algebraic system belongs to a finitely axiomatizable locally finite variety.Research supported by National Science Foundation Grant No. DMS-8302295.Presented by George Gratzer.     

Definability in the lattice of equational theories of semigroups

We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T ^ϖ} is definable, whereT ^ϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable. The research of both authors was supported by National Science Foundation Grant No. DMS-8302295     

Interactions of the directed plasma from a cathodic arc withelectrodes and magnetic fields

Curved magnetic ducts are frequently used to remove macroscopic-sized droplets from the plasma stream of cathodic vacuum arcs. The plasma of a cathodic vacuum arc in a magnetic filter is characterized by a strongly directional ion velocity (corresponding to 20-100 eV) and magnetized electrons. In the first section of this paper the effects of these features on the I-V characteristic curves of planar probes are identified and explained using a simple model. This is then used to interpret the interaction of the plasma with the walls of a biased quarter torus duct. Two small electrodes placed on the outer and inner sections of the curved duct wall show that the I-V characteristic is determined primarily by the electron-ion current balance at the wall on the outside of the curve. The application of a bias to a planar electrode on the outer wall section was found to give the same increase in throughput as a positive bias applied to the entire duct with the advantage of a much smaller electron current being drawn by the biasing power supply. The improvement in duct throughput achievable with positive-biasing of the duct wall was found to depend on both the configuration and strength of the magnetic field in the quarter torus filter. The plasma density profile and potential were unaffected by the application of the bias     

Measurement of the cascade time of ∑ in liquid helium

In a study of ∑⁻π⁺ production by stopped K⁻ in liquid helium, (19 ± 6)% of stopped ∑⁻ were observed to decay. This corresponds to a “cascade time” of 28 ± 8 psec.     

The Zig-Zag Property and Exponential Cancellation of Ordered Sets

This paper outlines a chapter in the theory of ordered sets that concerns decompositions of ordered sets using various natural operations introduced by Garrett Birkhoff in 1940 – chiefly, the operations of sum (disjoint union), Cartesian product, and exponentiation. This area has seen two brief, four-year flowerings of research activity during the past sixty-four years. Ivan Rival, working with several collaborators around 1978, obtained beautiful results on exponentiation which attracted other researchers to this topic. Altogether, about a dozen people contributed a chain of significant new results between 1978–1982. The subject then lay dormant until a major contribution by Jonathan D. Farley appeared in 1996. This pre-figured the solution, four years later, of a central problem about exponentiation of finite ordered sets that Birkhoff had formulated in 1942. The theory of operations on ordered sets is a beautiful chapter of mathematics. We concentrate on those portions that are most relevant for the solution of the Birkhoff problem, offering (hopefully) accessible proofs of many results, including some new results. We would like to see a new renaissance of activity in this area, which continues to offer many attractive open problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Toward an arithmetic of polynomials

Summary ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.